E xtinction and the O ptical theorem , B erg 9

Extinction and the Electromagnetic Optical Theorem

Matthew J.Berg, Christopher M. Sorensen, and Amit Chakrabarti

Kansas State University, Department of Physics, Manhattan, KS 66502-2601, USA tel: +1 785-317-3378, e-mail: [email protected]

Abstract

The energy flow caused by single particles is examined to form a physically based under- standing for extinction and the electromagnetic optical theorem. The behavior of the energy flow is explained in terms of wave interference. The optical theorem is shown to be a rela- tionship based on the relative phase between the wave incident on a particle and the wave scattered by it. The relative phase is determined by the refractive and absorptive properties of the particle and reveals how extinction depends on both scattering and absorption.

1 Introduction

The optical theorem has a long history; its occurrence in electromagnetic theory begins more than one hun- dred years ago and analogs of the theorem are found in quantum mechanical and acoustical scattering [1]. Over the years, many derivations and implementations of the theorem have been given [2]-[5]. However, we have found no previous work that provides an explicit interpretation of the theorem in a physical (not math- ematical) context. This work examines the optical theorem in detail and achieves a physical understanding of its meaning based on simulations of the energy flow caused by specific single particles.

2 The Optical Theorem

Consider a single particle that is illuminated by a plane wave polarized along the x-axis and traveling in the nˆ inc direction, see Fig. (1) and Eq. (2) below. By expanding this wave into counter-propagating spherical waves using Jones’ lemma, one can obtain an expression for the extinction cross section Cext of the particle as 4π Cext = xˆ · Esca nˆ inc . inc Im 1 ( ) (1) kEo This is the electromagnetic optical theorem [5]. To the unfamiliar investigator, Eq. (1) can appear mysterious. The optical theorem relates Cext to the sca inc imaginary part of the scattering amplitude E1 evaluated in only the forward direction nˆ . However, ex- tinction is due to scattering and absorption [3]. Scattering involves all directions whereas absorption is often independent of direction, so why should the optical theorem depend on only the forward direction, and how is absorption involved?

3 Theoretical Considerations

Suppose that the particle is located at the origin of the Cartesian coordinate system and surrounded by vacuum, see Fig. (1). The particle is described by a complex-valued refractive index m. Let S and V denote the surface and interior volume of the particle, respectively. The fields of the incident plane wave are k Einc(r) = Eincexp(ikrrˆ · nˆ inc) xˆ , Binc(r) = nˆ inc × Einc(r) . (2) o ω 10 Tenth C onference on L ight S cattering by N onspherical Particles

The wave number of the incident wave is k = 2π/λ where λ is the wavelength. A harmonic time dependence given by exp(−iωt), where ω is the angular frequency, is assumed for all field quantities. Surrounding the particle is a spherical surface Sl of radius Rl that is large enough that points on its surface are in the far-field of the particle. The intersection of Sl with the y-z plane forms the Cl contour, see Fig. (1). The wave scattered by the particle can be found by solving Maxwell’s equations using the volume integral relation [5]. In the far-field of the particle, the scattered wave takes the form of an outward traveling spherical wave with an sca angular profile given by the scattering amplitude E1 . The Poynting vector S describes the energy flow in an electromagnetic wave [6]. Because both the incident and scattered waves exist at the observation point r, S factors into three distinct terms. One of these terms, the cross term Scross, involves the fields of both the incident and scattered waves whereas the other terms involve the fields of only either the incident or scattered waves. The integral of the flow of the time- cross ext averaged cross term hS it through Sl gives the extinction cross section C , and, it is this integral that forms a starting point for the derivation of the optical theorem, Eq. (1) [5].

x

Figure 1: A scattering arrangement with a S{ star-shaped particle at the origin. The inci- Φ Einc dent wave is shown with the direction of its electric Einc and magnetic Binc fields point- r `inc ing along the x and y-axes respectively and n inc inc its propagation direction along nˆ . The B Θ observation point r is shown along with the S surface Sl of radius Rl and the intersection z of Sl with the y-z plane that forms the cir- C{ cular contour, C . y l

4 A Physical Picture of the Optical Theorem

cross Figure (2) reveals a major qualitative aspect of the behavior of the radial component of the hS it energy flow; the flow alternates from being inward to outward and does so more rapidly with direction as Rl in- cross creases. To see how integration of the hS it energy flow through Sl yields the extinction cross section, cross cross the integral I (θs) = hS it · rˆ da is also shown in Fig. (2). This integral is taken over the open sur- R∂Sl face ∂Sl which is formed by the part of Sl extending from θ = π to θ = θs. Because of energy conservation cross ext inc inc considerations, there is a negative sign in the relation I (θs = 0) = −C I , (where I is the flux of the ext ext inc incident wave), which is why the curves in Fig. (2) for I stop at the negative of C I when θs = 0. By expressing the scattered wave as a spherical wave with an amplitude and phase shift given by the sca scattering amplitude E1 , we demonstrate that the optical theorem can be understood in a more ‘physical’ setting than is done in existing literature. The ‘physical’ understanding relies on the interpretation of ex- tinction as being caused by the interference of the incident wave with the scattered wave. This interference causes the alternating energy flow shown in Fig. (2) and ultimately accounts for the value of Cext via a phase shift relationship between the two waves. This phase depends, in part, on the wave inside of the particle. As the refractive (Re m), or the absorptive (Im m) properties of the particle vary, the internal wave changes which effects the phase of the scattered wave and hence forms the connection between extinction and the optical properties of the particle. In addition, the energy-flow interpretation of the optical theorem naturally E xtinction and the O ptical theorem , B erg 11

cross Figure 2: Left column: Polar plots of the radial component of the hS it energy flow through the Cl contour shown in Fig. (1). The particle is a sphere with a size parameter kR = 4.08 and refractive index m = 1.10 + 0i and the scattered wave is calculated from Mie theory. The three polar plots labeled a, b and c show the energy flow for increasing contour radii Rl = 10R, Rl = 20R and Rl = 50R, respectively. Right column: Plots of the integral Icross for the contour radii corresponding to the matching polar plots in the left-hand column. The value of Cext as calculated directly from Mie theory is indicated in the plots. leads one to an understanding of the requirement regarding the size of a detector that is used to measure extinction, as discussed in Ref. [5]. 12 Tenth C onference on L ight S cattering by N onspherical Particles

Acknowledgments This work was supported by the NASA GSRP.

References

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[4] Jackson, J. D., Classical Electrodynamics (Wiley, New York, 1999).

[5] Mishchenko, M. I., Travis, L. D. and Lacis A. A., Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2002).

[6] Rothwell, E. J. and Cloud, M. J., Electromagnetics (CRC Press, New York, 2001).